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Theorem cantnflem1c 8182
Description: Lemma for cantnf 8188. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapval.f  |-  ( ph  ->  F  e.  S )
oemapval.g  |-  ( ph  ->  G  e.  S )
oemapvali.r  |-  ( ph  ->  F T G )
oemapvali.x  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1.o  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
Assertion
Ref Expression
cantnflem1c  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( G supp  (/) ) )
Distinct variable groups:    u, c, w, x, y, z, B    A, c, u, w, x, y, z    T, c, u    u, F, w, x, y, z    S, c, u, x, y, z    G, c, u, w, x, y, z    u, O, w, x, y, z    ph, u, x, y, z   
u, X, w, x, y, z    F, c    ph, c
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)    O( c)    X( c)

Proof of Theorem cantnflem1c
StepHypRef Expression
1 cantnfs.b . . 3  |-  ( ph  ->  B  e.  On )
21ad3antrrr 734 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  B  e.  On )
3 simplr 760 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  B )
4 oemapval.g . . . . . 6  |-  ( ph  ->  G  e.  S )
5 cantnfs.s . . . . . . 7  |-  S  =  dom  ( A CNF  B
)
6 cantnfs.a . . . . . . 7  |-  ( ph  ->  A  e.  On )
75, 6, 1cantnfs 8161 . . . . . 6  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
84, 7mpbid 213 . . . . 5  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
98simpld 460 . . . 4  |-  ( ph  ->  G : B --> A )
10 ffn 5737 . . . 4  |-  ( G : B --> A  ->  G  Fn  B )
119, 10syl 17 . . 3  |-  ( ph  ->  G  Fn  B )
1211ad3antrrr 734 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  G  Fn  B )
13 oemapval.t . . . . . . 7  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
14 oemapval.f . . . . . . 7  |-  ( ph  ->  F  e.  S )
15 oemapvali.r . . . . . . 7  |-  ( ph  ->  F T G )
16 oemapvali.x . . . . . . 7  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
175, 6, 1, 13, 14, 4, 15, 16oemapvali 8179 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
1817simp3d 1019 . . . . 5  |-  ( ph  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
1918ad3antrrr 734 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
20 cantnflem1.o . . . . . . 7  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
215, 6, 1, 13, 14, 4, 15, 16, 20cantnflem1b 8181 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
2221ad2antrr 730 . . . . 5  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  C_  ( O `  u ) )
23 simprr 764 . . . . 5  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( O `  u
)  e.  x )
2417simp1d 1017 . . . . . . . 8  |-  ( ph  ->  X  e.  B )
25 onelon 5458 . . . . . . . 8  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
261, 24, 25syl2anc 665 . . . . . . 7  |-  ( ph  ->  X  e.  On )
2726ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  e.  On )
28 onss 6622 . . . . . . . . . 10  |-  ( B  e.  On  ->  B  C_  On )
291, 28syl 17 . . . . . . . . 9  |-  ( ph  ->  B  C_  On )
3029sselda 3461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  On )
3130adantlr 719 . . . . . . 7  |-  ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  ->  x  e.  On )
3231adantr 466 . . . . . 6  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  On )
33 ontr2 5480 . . . . . 6  |-  ( ( X  e.  On  /\  x  e.  On )  ->  ( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
3427, 32, 33syl2anc 665 . . . . 5  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
3522, 23, 34mp2and 683 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  e.  x )
36 eleq2 2493 . . . . . 6  |-  ( w  =  x  ->  ( X  e.  w  <->  X  e.  x ) )
37 fveq2 5872 . . . . . . 7  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
38 fveq2 5872 . . . . . . 7  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
3937, 38eqeq12d 2442 . . . . . 6  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
4036, 39imbi12d 321 . . . . 5  |-  ( w  =  x  ->  (
( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  <-> 
( X  e.  x  ->  ( F `  x
)  =  ( G `
 x ) ) ) )
4140rspcv 3175 . . . 4  |-  ( x  e.  B  ->  ( A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  ->  ( X  e.  x  ->  ( F `  x )  =  ( G `  x ) ) ) )
423, 19, 35, 41syl3c 63 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =  ( G `
 x ) )
43 simprl 762 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =/=  (/) )
4442, 43eqnetrrd 2716 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  =/=  (/) )
45 fvn0elsupp 6932 . 2  |-  ( ( ( B  e.  On  /\  x  e.  B )  /\  ( G  Fn  B  /\  ( G `  x )  =/=  (/) ) )  ->  x  e.  ( G supp  (/) ) )
462, 3, 12, 44, 45syl22anc 1265 1  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( G supp  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777    C_ wss 3433   (/)c0 3758   U.cuni 4213   class class class wbr 4417   {copab 4474    _E cep 4754   `'ccnv 4844   dom cdm 4845   Oncon0 5433   suc csuc 5435    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   supp csupp 6916   finSupp cfsupp 7880  OrdIsocoi 8015   CNF ccnf 8156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-seqom 7164  df-1o 7181  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-oi 8016  df-cnf 8157
This theorem is referenced by:  cantnflem1  8184
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